Chapter 12 Probability © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 12: Probability 12.1 Basic Concepts 12.2 Events Involving “Not” and “Or” 12.3 Conditional Probability; Events Involving “And” 12.4 Binomial Probability 12.5Expected Value

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 12-1 Basic Concepts

© 2008 Pearson Addison-Wesley. All rights reserved Basic Concepts Historical Background Probability The Law of Large Numbers Probability in Genetics Odds

© 2008 Pearson Addison-Wesley. All rights reserved Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling was Pierre Simon de Laplace, who is often credited with being the “father” of probability theory. In the twentieth century a coherent mathematical theory of probability was developed through people such as Chebyshev, Markov, and Kolmogorov.

© 2008 Pearson Addison-Wesley. All rights reserved Probability The study of probability is concerned with random phenomena. Even though we cannot be certain whether a given result will occur, we often can obtain a good measure of its likelihood, or probability.

© 2008 Pearson Addison-Wesley. All rights reserved Probability In the study of probability, any observation, or measurement, of a random phenomenon is an experiment. The possible results of the experiment are called outcomes, and the set of all possible outcomes is called the sample space. Usually we are interested in some particular collection of the possible outcomes. Any such subset of the sample space is called an event.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Tossing a Coin If a single fair coin is tossed, find the probability that it will land heads up. Solution The sample space S = {h, t}, and the event whose probability we seek is E = {h}. P(heads) = P(E) = 1/2. Since no coin flipping was actually involved, the desired probability was obtained theoretically.

© 2008 Pearson Addison-Wesley. All rights reserved Theoretical Probability Formula If all outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by

© 2008 Pearson Addison-Wesley. All rights reserved Example: Flipping a Cup Solution A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. Find the probability that it will land on its top. From the experiment it appears that P(top) = 10/100 = 1/10. This is an example of experimental, or empirical probability.

© 2008 Pearson Addison-Wesley. All rights reserved Empirical Probability Formula If E is an event that may happen when an experiment is performed, then the empirical probability of event E is given by

© 2008 Pearson Addison-Wesley. All rights reserved Example: Card Hands Solution There are 2,598,960 possible hands in poker. If there are 36 possible ways to have a straight flush, find the probability of being dealt a straight flush.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Gender of a Student Solution A school has 820 male students and 835 female students. If a student from the school is selected at random, what is the probability that the student would be a female?

© 2008 Pearson Addison-Wesley. All rights reserved The Law of Large Numbers As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular event will tend to come closer and closer to the theoretical probability of that event.

© 2008 Pearson Addison-Wesley. All rights reserved Probability in Genetics Gregor Mendel, an Austrian monk used the idea of randomness to establish the study of genetics. To study the flower color of certain pea plants he found that: Pure red crossed with pure white produces red. Mendel theorized that red is “dominant” (symbolized by R), while white is recessive (symbolized by r). The pure red parent carried only genes for red (R), and the pure white parent carried only genes for white (r).

© 2008 Pearson Addison-Wesley. All rights reserved Probability in Genetics Every offspring receives one gene from each parent which leads to the tables below. Every second generation is red because R is dominant. Second Parent rr First Parent RRr R 1 st to 2 nd Generation Second Parent Rr First Parent RRRRr rrRrr 2 nd to 3 rd Generation offspring

© 2008 Pearson Addison-Wesley. All rights reserved Example: Probability of Flower Color Referring to the 2 nd to 3 rd generation table (previous slide), determine the probability that a third generation will be a) redb) white Base the probability on the sample space of equally likely outcomes: S = {RR, Rr, rR, rr}.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Probability of Flower Color Solution a) Since red dominates white, any combination with R will be red. Three out of four have an R, so P(red) = 3/4. b) Only one combination rr has no gene for red, so P(white) = 1/4.

© 2008 Pearson Addison-Wesley. All rights reserved Odds Odds compare the number of favorable outcomes to the number of unfavorable outcomes. Odds are commonly quoted in horse racing, lotteries, and most other gambling situations.

© 2008 Pearson Addison-Wesley. All rights reserved Odds If all outcomes in a sample space are equally likely, a of them are favorable to the event E, and the remaining b outcomes are unfavorable to E, then the odds in favor of E are a to b, and the odds against E are b to a.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Odds 200 tickets were sold for a drawing to win a new television. If Matt purchased 10 of the tickets, what are the odds in favor of Matt’s winning the television? Solution Matt has 10 chances to win and 190 chances to lose. The odds in favor of winning are 10 to 190, or 1 to 19.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Converting Probability to Odds Suppose the probability of rain today is.43. Give this information in terms of odds. Solution We can say that 43 out of 100 outcomes are favorable, so 100 – 43 = 57 are unfavorable. The odds in favor of rain are 43 to 57 and the odds against rain are 57 to 43.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Converting Odds to Probability Your odds of completing College Algebra class are 16 to 9. What is the probability that you will complete the class? There are 16 favorable outcomes and 9 unfavorable. This gives 25 possible outcomes. So Solution